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A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs

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arxiv 1310.0103 v3 pith:5DAZUD6Z submitted 2013-10-01 math.RT math.QA

A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs

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keywords typequantumtheorykazhdan-lusztigsymmetricapproachcoidealgroup
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We show that Hecke algebra of type B and a coideal subalgebra of the type A quantum group satisfy a double centralizer property, generalizing the Schur-Jimbo duality in type A. The quantum group of type A and its coideal subalgebra form a quantum symmetric pair. A new theory of canonical bases arising from quantum symmetric pairs is initiated. It is then applied to formulate and establish for the first time a Kazhdan-Lusztig theory for the BGG category O of the ortho-symplectic Lie superalgebras $\mathfrak{osp}(2m+1|2n)$. In particular, our approach provides a new formulation of the Kazhdan-Lusztig theory for Lie algebras of type B/C.

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